# An adelic arithmeticity theorem for lattices in products

## TL;DR

This paper establishes an adelic arithmeticity theorem for lattices in products of semi-simple Lie groups and totally disconnected locally compact groups, broadening the understanding of their arithmetic nature without finiteness or compact generation assumptions.

## Contribution

It proves that such lattices are, under mild conditions, arithmetic, even without assumptions of finite generation or compactness of the ambient group.

## Key findings

- Lattices in these product groups are arithmetic under mild assumptions.
- The theorem applies without requiring the lattice to be finitely generated.
- The ambient group need not be compactly generated.

## Abstract

We prove that, under mild assumptions, a lattice in a product of semi-simple Lie group and a totally disconnected locally compact group is, in a certain sense, arithmetic. We do not assume the lattice to be finitely generated or the ambient group to be compactly generated.